2022/10/25 by Jamie M. Taylor, David Pardo, Taylor, Jamie M. +3 · 3 citations
Engineering · Physics and Astronomy · #Advanced Numerical Analysis Techniques #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2210.14129
openalex publication_date 2022/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
When using Neural Networks as trial functions to numerically solve PDEs, a key choice to be made is the loss function to be minimised, which should ideally correspond to a norm of the error. In multiple problems, this error norm coincides with--or is equivalent to--the H-1-norm of the residual; however, it is often difficult to accurately compute it. This work assumes rectangular domains and proposes the use of a Discrete Sine/Cosine Transform to accurately and efficiently compute the H-1 norm. The resulting Deep Fourier-based Residual (DFR) method efficiently and accurately approximate solutions to PDEs. This is particularly useful when solutions lack H2 regularity and methods involving strong formulations of the PDE fail. We observe that the H1-error is highly correlated with the discretised loss during training, which permits accurate error estimation via the loss.